Equivariant surgery theories and their periodicity properties
The book under review is an exposition and survey of certain formal properties of equivariant surgery theories. Before giving more details about the contents of the book, it might be useful to briefly describe this subject. Equivariant surgery is concerned with the study of manifolds with group action up to variation by bordisms constructed from ``elementary surgeries''. If N is a compact n-dimensional smooth G-manifold, let \(N^ H\) denote the fixed-point set of a subgroup \(H\subseteq G\). For convenience, I will assume that G is a finite group. Let \(f_ 1: S^ k\times D^{n-k}\to N\) be a framed embedding such that (i) \(f_ 1\) is an H-equivariant smooth map, (ii) \(f_ 1(S^ k\times 0)\subset N^ H\), and (iii) the image \(f_ 1(S^ k\times D^{n-k})\) is a smooth H- tubular neighbourhood of \(f_ 1(S^ k\times 0)\) in N. These properties fix the H-action on \(S^ k\times D^{n-k}\) as the product of the trivial action on the first factor and a linear H-representation on the second factor. If we also assume that the image of \(f_ 1\) is disjoint from all its translates by elements of G-H, then \(f_ 1\) extends to a G- embedding \[ f:\coprod_{\alpha \in G/H}(S^ k\times D^{n- k})_{\alpha}\to N \] induced by the action of G. The basic operation in equivariant surgery is to construct the bordism \[ W=(N\times I)\cup_ f\coprod_{\alpha \in G/H}(D^{k+1}\times D^{n-k})_{\alpha}, \] referred to as the trace of an elementary surgery on the class \(f_ 1(S^ k\times 0)\). The manifold \(N'\) appearing as the ``top'' boundary of W is the result of performing equivariant surgery on the given class in N. There is a map F: \(W\to N\) extending the identity on \(N\times 0\subset W\) which maps the equivariant handle to a G-orbit of points in N. More generally, if we start with a G-map h: \(N\to M\), where M is another G-manifold of the same dimension, then this construction provides a bordism between the maps h and \(h': N'\to M.\) There are various natural questions connected with this construction. First, given a map h: \(N\to M\), how can we decide which G-surgeries are possible? Second, the process suggests that we should identify N and M if there exists h: \(N\to M\) which is ``optimal'' from the point of view of G-surgery. What is an appropriate definition of ``optimal''? The set of equivalence classes with respect to a chosen optimality property is called the structure set of M and denoted \({\mathcal S}(M).\) Fortunately the non-equivariant case \((G=1)\) was fully developed before these questions where first seriously considered by T. Petrie in the mid 1970's. The surgery theory of Browder, Milnor, Novikov, Sullivan, Thom, and Wall gives a guide for equivariant case as well. For the first question, the answer is to assume that h has degree one and comes with some form of ``bundle data''. The most natural way to define this is to assume that there is a G-bundle \(\xi\) over M and an ``\({\mathbb{R}}\)-stable'' G-bundle map \(\hat h:\) \(\nu\) \(N\to \xi\) covering h. Such maps are called normal maps in the theory. This is the approach of Lück and Madsen, which differs somewhat in formulation from the original approach of Petrie. The second question is more difficult, since the equivariant setting allows more choices. The main options seem to be to aim for either (i) equivariant homotopy equivalences or (ii) equivariant maps which are homotopy equivalences (so-called ``pseudo-equivalences''). The difference is that a pseudo-equivalence need not have a homotopy inverse which is a G-map. These different choices lead to different theories, with their own areas of application to geometric problems. Once the basic definitions are settled, the main tasks of the theory are to (i) describe the set \({\mathcal N}(M)\) of normal maps with target M in homotopy-theoretic terms, and (ii) to compute the structure set by analyzing the natural map \({\mathcal S}(M)\to {\mathcal N}(M)\). The second part leads to the definition and study of the equivariant surgery obstruction groups, whose properties are the main subject of the book under review. The full relationship among these three objects, structure set, normal map set and surgery obstruction groups is expressed in the ``surgery exact sequence''. Most of the work done so far on the development of the theory is based on the philosophy that one should deal with equivariant surgery stratum by stratum, with respect to the stratification of M by fixed sets of subgroups of G. This leads either to stronger assumptions on the normal maps (as for example using isovariant maps in the Browder- Quinn theory) or to stronger assumptions on the normal maps (as for example using isovariant maps in the Browder-Quinn theory) or to stronger assumptions on the underlying manifolds. The most usual form for the latter is the ``Gap Hypothesis'', which states that each stratum is codim\(\geq 3\) in any stratum containing it. With this background, we can now appreciate the goals and results of the present book. First, in Chapters 1, 2 the authors describe and compare the various equivariant surgery theories, with emphasis on their formal properties such as change of coefficients, restriction to subgroups, and existence of surgery exact sequences. Chapter 3 contains new material on periodicity properties of the surgery obstruction groups for surgery up to G-homotopy equivalence. The basic idea is to study the products given crossing a surgery problem h: \(N\to M\) with another G-manifold P in domain and range. For certain ``periodicity manifolds'' P the resulting surgery obstruction groups for M and \(M\times P\) are isomorphic. In Chapter 4, further product pairings are defined following the work of Yoshida. These lead to periodicity for equivariant surgery with coefficients. Finally in Chapter 5, the analogues of products and periodicity are studied for surgery up to pseudo-equivalence. For the main conclusions, the group G is assumed to be nilpotent of odd order. Although the subject is very technical, the authors have worked hard to present the material in a clear and direct manner. The comparison of the various equivariant theories is particularly valuable. In addition, the new results in the book leave the theory much better off than before, with many new tools for computation of specific geometric examples.
stratification by fixed sets of subgroups
equivariant maps which are homotopy equivalences
periodicity properties of the surgery obstruction groups
scientific article; zbMATH DE number 47878